A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

Find out what a "fault-free" rectangle is and try to make some of your own.

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

In this matching game, you have to decide how long different events take.

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Can you find all the different ways of lining up these Cuisenaire rods?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

A Sudoku that uses transformations as supporting clues.

Using the statements, can you work out how many of each type of rabbit there are in these pens?

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

There are lots of different methods to find out what the shapes are worth - how many can you find?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

What is the smallest number of coins needed to make up 12 dollars and 83 cents?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

A few extra challenges set by some young NRICH members.

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Investigate the different ways you could split up these rooms so that you have double the number.